ZEROTH DAY

On AI and pure mathematics

Is artificial intelligence on track to meet the expectations of its investors, who just in 2020 poured $50 billion into the industry?  AI's record of missed deadlines for predicted milestones is as old as its name.  But literary production on the subject could hardly be more extensive.  Reading all the non-technical books on my local bookstore's AI shelf would be more than a full-time job, leaving less than no time for my real job, which AI has not yet eliminated.  Even the sub- or parallel discipline of AI ethics now occupies 10 pages of footnotes on the English-language Wikipedia page and 1400 pages published in the last two years by Oxford University Press, on my own bookshelf; practically every day I discover another 100 pages or so.   I have nevertheless forced myself to dip into a representative sample as preparation for an experiment that is beginning to take shape with this text. 

Most of what I've read tries to address the question of just how "intelligent" the products of this industry have been up to now, or will be in the near future, or what it would take for actually existing AI to deserve to be called "intelligent," or whether it would be a good thing, or whether it's even possible.  None of these is my problem.  Or rather, they are my problem, but only as a citizen of my country, or of borderless civilization, concerned, like everyone else, by what the massive implementation of ostensibly intelligent artificial systems would entail for what matters to me — not least, whether it would make sense for these things to continue to matter to me, or perhaps more accurately whether what matters to me would still matter to anyone or anything else, if the ambitions of AI's promoters even minimally come to fruition.

Whether or not these ambitions are realistic or whether the AI industry has produced anything genuinely intelligent — two questions I am thoroughly unqualified to answer — it certainly has produced perceptible changes in the social, economic, professional, and political lives of my contemporaries, and in so doing it has had concomitant effects on the ideas we use to think about our social, economic, professional, and political lives.   My motivation in undertaking this experiment is to understand the consequences of this way of thinking for my own vocation of pure mathematics, which is marginal to the concerns of most of those at risk of the AI project's collateral damage but which has been central to the project's imagination and its aspirations from the very outset. 

It is possible to view the growing interest in automated proof verification and artificial theorem proving, two aspects of a still largely hypothetical AI future of mathematics, as stemming from purely internal factors that govern the profession's development as it evolves to meet its autonomously defined goals.  The ideal of incontrovertible proof has been bound up with mechanization since it was first articulated, and the logic that ultimately made digital computers possible is a direct outgrowth of the attempt to perfect this ideal in the development of symbolic and philosophical logic in the late 19th and early 20th century, and can even be seen as a byproduct of the proof of the absolute impossibility of realizing this ideal. 

I don't think this view is plausible, given the saturation of our culture with AI themes and memes, that goes well beyond bookstores' overloaded AI shelves.  Just look, for Godzilla's sake, at the blockbuster film that a few months ago was supposed to signal the beginning of post-pandemic normality.   It was basically the story of competing forms of natural violence uniting to stop a runaway AI; the only plot element that didn't look entirely computer-generated was a not entirely unsympathetic treatment of AI paranoia.

Even my limited immersion in this literature, and in the cultural moment, has placed me in a bind.  I want to write about mathematics, at most allowing myself detours through politics and ethics and the analysis of texts in order to understand how the words and power behind AI and their interrelations are shaping the future of my vocation and its values.   But the literature on AI has grown so tangled that I can hardly claim to have understood its scope without additional detours through neurology, cognitive psychology, linguistics, digital humanities, finance,  advertising, and much much more, not to mention the disciplines like robotics and computer science that are central to the AI project but that I long ago made the conscious decision to keep at arm's length, and that I pledge never to claim to understand.

This post is meant to be the first of a series of texts exploring the reasons for the absence of any sustained discussion of these issues on the part of mathematicians, in contrast to the very visible public debate about the perils and promises of AI.  Much of my book Mathematics without Apologies was devoted to a critique of claims regarding the "usefulness" of mathematics when, as is nearly always the case, they are not accompanied by close examination of the perspectives in which an application of mathematics may or may not be seen as "useful."  Similarity with the intended critique of the uncritical use of words (like "progress") that accompany the ideology surrounding mechanization — mechanical proof verification and automated theorem proving, in particular — will be apparent.  The reason should be obvious:  unless we can conceive an alternative to conventional measures of utility for which human mathematics is a positive good, the forces that make decisions about this sort of thing will declare my vocation obsolete.  Most of my colleagues who are involved in advancing the mechanization program have conceded the rhetorical battle and some are already forecasting the demise of human mathematics.  So the plan is to continue the discussion in this new format, and gradually to phase out the blog that I launched when Mathematics without Apologies was published, as I have already tried and failed to do once before. 

Because I will be forced to draw on so many different disciplinary perspectives in the course of exploring the topic of mechanization, there is a real danger that these texts will lose any chance of forming a coherent whole.  For my own sake, then, as much as for the sake of potential readers, I propose a slogan that is meant to hold everything together until I come up with a better slogan.  Here it is: 

Current trends in mechanization belong to the history of mathematics, both as events in a historical process and in the creation of common narratives about the meaning of the process. 

I am borrowing a point about historiography made by Michel-Rolph Trouillot in Silencing the Past:

Not only can history mean either the sociohistorical process or our knowledge of that process, but the boundary between the two meanings is often quite fluid.

The significance for mathematics is this:   Practically everything that mathematicians have written about the mechanization project takes the position that it represents a way to make mathematics "better" at achieving its goals.  However, like every historical process, it is simultaneously a process of redefining these goals.  The stakes, then, of mechanization, are what I will call existential:  they have to do with the nature or self-understanding of mathematics.  It shouldn't be necessary to say that it this not at all a settled matter. 

More simply:  the question of whether or not mechanization will make mathematics "better" is literally nonsensical.  This is because, if mechanization brings about changes as radical as what its proponents claim, then "mathematics" will mean something radically different at the end of the process from what it meant at the beginning.  That's not to say that it's nonsensical to ask whether or not mechanization is a good thing.  It just means that the the question can't be answered solely in reference to mathematics as currently understood.   

A recent article by Paul Ernest makes this point in another way:

There is a fundamental ambiguity at the heart of the question as to what constitutes good mathematics. From an ethical perspective good pure mathematics is that mathematics which benefits humankind and contributes to human flourishing. From an epistemic perspective good mathematics is that pure mathematics which is clearly expressed, which is well justified, normally by means of mathematical proofs, and which generally conforms well to the epistemic standards of the social practice of mathematics and of the community of mathematicians.

Ernest illustrates the distinction in the setting of applied mathematics: 

In the case of applied mathematics ethically good applications are those that are beneficial to humankind and cause little or no harm. Applications that are good from an epistemic perspective are those that are correctly and rigorously formulated and that accurately predict outcomes or provide good explanatory models for their target domains.

But how can pure mathematics, with no view to applications, benefit humankind and contribute to human flourishing?   Ernest's text seeks — and his path is by no means straightforward — to reconcile a commitment to the "internal goods" of mathematics (in the sense of Alasdair MacIntyre's virtue ethics), the ethical responsibility to the social practice of mathematics with the ethical imperative to be a "virtuous person," a responsible actor within society. 

That is also the ambition of this series of texts, specifically as a response to the challenges posed by mechanization, which promises to improve mathematics consistently with (one version of) the epistemic perspective.  I will be arguing that it in fact promises to substitute a new epistemic perspective in which expressions like "social practice" and "community of mathematicians" are at best optional.  Its implications for the ethical perspective must have been largely neglected, both by its partisans and by its opponents; otherwise, how can one imagine that the prospect of replacing human mathematicians by machine, whether evoked in fear or in jest, will "contribute to human flourishing"?

Meredith Broussard reminds us not to neglect the ethical perspective at the end of Artificial Unintelligence:

Turning real life into math is a marvelous magic trick, but too often the inconveniently human part of the equation gets pushed to the side. Humans are not now, nor have they ever been, inconvenient. Humans are the point. Humans are the beings all this technology is supposed to serve.

Broussard apparently suspects (as do I) that not all the developers and promoters and publicists of AI agree that "humans are the point"; otherwise why would she find it necessary to say so?

Now for a few ground rules that will govern this experiment.

I will not predict that mathematicians will never decide en masse to submit to the discipline of automated proof checkers, nor even that mathematics will be downgraded from a profession to a pastime, because I am not a fortune-teller.

I will not say things like "this [fill in the blank] is not mathematics."  Instead, I will share my thoughts on what it means to say and on who is entitled to say "this is" or "this is not mathematics," and I will attempt to track how the consensus regarding the boundaries of what qualifies as mathematics has shifted over time.

I will not claim familiarity with any of the formal systems used in the design of automated proof checkers, nor to understand any of the software that implements the actual automatic verification, much less to understand the details of current or future work on AI, whether or not it is applied to mathematics.  Even when I have a pretty good idea of what is going on with some of these systems, I will fiercely deny any technical understanding whatsoever, because my understanding of the technicalities should never be an issue. 

I will not attempt to write polished texts, otherwise I won't ever get started.

You should see these as pledges or promises, and if you believe you have caught me writing or doing something I promised not to do, please check that you are not reading your expectations rather than my words.

On the other hand,

I will make a point of presenting positions with which I disagree, especially those with which I disagree most vigorously, in their strongest form.  I can't promise I'll have time to respond adequately to comments so I will (probably) not allow them on this platform, but I will try to find a way, separate from Substack, to allow readers to tell me when they think I have been unfair or simply obtuse.

I will make every effort to illustrate the occasional substantive claims with examples taken from mathematics, drawing on the largest possible range of subspecialties, historical periods, and cultural traditions. 

I will punctuate these essays with regular reviews of the books and articles I am reading that are related more or less directly to my main topic.  This has the advantage, for me, of helping me to register what I have read; and for you, of relieving the monotony of my authorial voice and my philosophical and ethical obsessions.

A few words are in order about my decision to move my blog to Substack, in spite of its origins in efforts to forge a "new narrative" for tech — and in spite of its name that, for a mathematician, is terribly misleading.  The most basic reason is that, as I noted in 2016, my old blog had by then already "reached the end of its natural life" as a disorganized 120,000-word appendage to my book.  This means concretely that the only way to squelch the temptation to relitigate the controversies that my book may have raised and that remained unsettled, as controversies tend to do — to have the last word (and since 2016 my blog has published more like 40,000 further last words) — is to declare them stalemated, and to start thinking about something else. 

I was already thinking obsessively about the project to mechanize mathematics long before I wrote an essay on the topic, and I have been jotting down scattered thoughts about mechanization and selecting some of them for public lectures.  Writing regularly in a new format, without the distractions entailed by defending the opinions in my book, will allow me to develop the loose ends and to organize the results into something resembling a coherent thesis.  And I can hope that the prospect of devoting myself to a regular writing program will free me from the rut that swallowed up my prose style as well as my sunny disposition at the onset of the pandemic.

A second reason is that Substack has (reportedly) more than 12 million monthly visits and (at last count) more than 500,000 paid subscribers, along with thousands of regular authors.  But as far as I can tell (and it may just be that I don't know where to look), only one of Substack's newsletters is specifically devoted to mathematics.   Is that limit inherent in the medium or is there room to grow?  Testing the medium is the only way to find out.   

Finally, as you may have read, Substack has been a source of steady income for a handful of lucky authors, a bitter disappointment for a much greater number of contributors, and a goldmine for the platform's creators.   I have confessed in public that the prospect of profiting materially from my writing habit is not altogether foreign to my calculations.  This was already the case even before two new friends tried to convince me to write a second book, one that would leave an impression on the market, and not only on the minds of its readers.  I tried, twice, getting as far as drafting book proposals.  At that point my well-intentioned friends decided I had better stick to my day job — which, as I already said, AI has not yet eliminated.  But rather than let the material I collected gather microelectronic dust on my hard drive, I can put it to use in an intellectual experiment that will double, inconspicuously, as a market survey.  For the time being I do not intend to exploit Substack's option to charge for subscriptions.  But that may change if the market survey points me in a different direction.

I have been toying with the idea of working out my issues with mathematical mechanization on Substack ever since last October, when I discovered the platform by subscribing to one of its newsletters (no reward for correctly guessing which one).  This means that for more than nine months I have been mulling over the implications of believing that mechanization is a part of the future of mathematics — I have no problem with this belief — and whether or not this belief inevitably entails that mechanization is the future of mathematics.  Every thought experiment along these lines has led me unexpectedly but inexorably to the question, "Why do anything at all?"  from which it is only a short hop to the question, "Why people?" 

I'll try to keep allusions to these questions to a minimum.   But they can't be avoided entirely, because most arguments for AI, taken to their logical conclusions, do seem to imply — contra Meredith Broussard — that humanity is getting close to its use-by date.  You will encounter some real live enthusiastic proponents of this idea when I get around to reviewing Mark O'Connell's To Be a Machine.  More disturbing is the persistent claim that AI will eventually be "better" than humans at anything you can name.  Computer scientist Scott Aaronson, for example, recently wrote this on the Scientific American blog:

It’s conceivable that someday, computers will replace humans at all aspects of mathematical research — but it’s also conceivable that, by the time they can do that, they’ll be able to replace humans at music and science journalism and everything else!

Aaronson was expressing skepticism about prospects for rapid digitization of mathematics, but his comment can also be read as a prophecy.

Most of all I do not want to contribute to promoting any ideas, merely by alluding to them, that would lead even a single actual human being to despair of existence.  And I think that the prophets of mechanized mathematics should be less relaxed about this danger.  No fewer than four of my fellow undergraduate and graduate students in mathematics had committed suicide by the time I completed my own Ph.D., and mathematical suicides continue, though at a much slower pace in my own circles.  Mathematicians are sensitive creatures.  If you only take away one thing from this newsletter, that's what I want you to remember.

Comments are disabled on this site, but readers are invited to submit comments to the author’s blog. Not all comments will be published.